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Efficient frequency-domain Full Waveform Inversion (FWI) oflong-offset/wide-azimuth node data can be designed with a few discretefrequencies. However, 3D frequency-domain seismic modeling remains challengingsince it requires solving a large and sparse linear indefinite system perfrequency. When such systems are solved with direct methods or hybriddirect/iterative solvers, based upon domain decomposition preconditioner,finite-difference stencils on regular Cartesian grids should be designed toconciliate compactness and accuracy, the former being necessary to mitigate thefill-in induced by the Lower-Upper (LU) factorization. Compactness isclassically implemented by combining several second-order accurate stencilscovering the eight cells surrounding the collocation point, leading to theso-called 27-point stencil. Accuracy is obtained by applying optimal weights onthe different stiffness and consistent mass matrices such that numericaldispersion is jointly minimized for several number of grid points perwavelength ($G$). However, with this approach, the same weights are used ateach collocation point, leading to suboptimal accuracy in heterogeneous media.In this study, we propose a straightforward recipe to improve the accuracy ofthe 27-point stencil. First, we finely tabulate the values of $G$ covering therange of wavelengths spanned by the subsurface model and the frequency. Then,we estimate with a classical dispersion analysis in homogeneous media thecorresponding table of optimal weights that minimize dispersion for each $G$treated separately. We however apply a Tikhonov regularization to guaranteesmooth variation of the weights with $G$. Finally, we build the impedancematrix by selecting the optimal weights at each collocation point according tothe local wavelength, hence leading to a wavelength-adaptive stencil.

Accurate 3D frequency-domain seismic wave modeling with the  wavelength-adaptive 27-point finite-difference stencil: a tool for full  waveform inversion